Write an absolute value inequality that has solutions (-2, 3) U (7, 12).

Answer 1

5 is halfway between 3 and 7. 5 is 2 units from 3 and 7. 5 is 7 units from -2 and 12.

An inequality with the given solution set is: 2 < l x - 5 l < 7

Case 1: x - 5 > 0

Then, 2 < x - 5 < 7. So, 7 < x < 12. In other words, x is in the interval (7,12).

Case 2: x - 5 < 0

Then, 2 < -x + 5 < 7. So, -3 < -x < 2. Therefore, 3 > x > -2. That is, x is in the interval (-2,3).

Case 3: x - 5 = 0. Then 2 < 0 < 7, which is nonsense.